Discuss the relative merits of using a proportional plus integral controller over purely proportional control in modifying system performance? [5] A second-order type-0 system with transfer function

G(s)=1/(s+1)(s+5)

is to be compensated using the controller

Gc(s)=1+K/s

under unity feedback. For the overall feedback system:

(i) Derive the system characteristic equation [5]
(ii) Obtain the value of integral gain K for which the system becomes unstable [5]
(iii) Discuss how K can be designed to meet relative stability criteria.

To save energy, the air supply to a 2000 ft office has been shut off overnight and the room temperature has dropped to 40°f. in the morning, the thermostat is reset to 70°f and warm air at 120 f begins to flov in at 200ft'/min. the air is well mixed within the room, and an equal mass flow of air at room temperature (changing with time) is withdrawn through a return duct. the air pressure is nearly 1 atm everywhere. ignoring heat transfer with the surroundings and kinetic and potential energy effects, estimate how long it takes for the room temperature to reach 70°f. plot the room temperature as a function of time.